Math @ Duke

Publications [#348659] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL; Clelland, JN, Flat metrics with a prescribed derived coframing,
Symmetry, Integrability and Geometry: Methods and Applications, vol. 16
(January, 2020) [doi]
(last updated on 2021/05/10)
Abstract: The following problem is addressed: A 3manifold M is endowed with a triple Ω =(Ω , Ω , Ω ) of closed 2forms. One wants to construct a coframing ω =(ω , ω , ω ) of M such that, first, dω = Ω for i = 1, 2, 3, and, second, the Riemannian metric g = ( ) + (ω ) + (ω ) be flat. We show that, in the ‘nonsingular case’, i.e., when the three 2forms Ω p span at least a 2dimensional subspace of Λ (T *M) and are realanalytic in some pcentered coordinates, this problem is always solvable on a neighborhood of p (Formula Presented) M, with the general solution ω depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution ω can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when Ω , Ω , Ω are scalar multiples of a single 2form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied. 1 2 3 1 2 3 i i 1 2 2 2 3 2 i 2 1 2 3 ω p


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

